Our Pain Scale
My knees have been bothering me of the past bunch of years. Everytime I visit an orthopedic expert they ask questions like these. Is the pain acute? Is the pain localized? Is the pain symmetric? Can you quantify the pain? This last question is interesting, because it opens up a Pandora's Box about relative pain and some sort of "measure" of the severity of that pain. Often, we are trying to be objective about a very subjective topic in coming up with a number between - say - 1 and 10. For what it is worth, here is my pain scale regarding my knees when I am out on a hike through the woods.
- Barely noticeable fleeting pain.
- Minimal persistent pain.
- Enough discomfort to change my gait.
- Enough discomfort to change my intended route.
- Enough discomfort to stay home and ice my knee(s).
- Enough discomfort to seek pain relief.
- Enough discomfort to call a doctor.
- Enough discomfort to take a nap.
- Reverting to the fetal position.
- I have now passed out.
Everyone has a different pain scale and there are no right or wrong answers when addressing these questions. From what I gather, elite military personnel get special training in pain management.
Our Risk Tolerance
The pain scale is a quantification of a physical stimulus when something bad happens to our bodies. Our risk tolerance (RT) is a measure of our capacity and willingness to endure financial "pain" as it applies to our investment portfolios. The physical symptoms of this financial pain can include (from bad to worse): distraction from our everyday lives, inability to sleep at night, and doing the exact "wrong" thing at the exact "right" time. We humans have an amazing knack of throwing in the towel at the precise moment we need to dry ourselves off after taking a bath. Just like a pain scale, we can arbitrarily choose a risk tolerance scale of 0 to 10. The investor with a 0 tolerance is intolerant of any losses to their investments over a certain period of time while the person with a 10 is very tolerant of similar losses.
Our Risk Aversion
If we define risk aversion (RA) as the opposite of risk tolerance, we can model this as our risk tolerance subtracted from an arbitrary constant. To make the math simple, let us just again choose the number 10 as our constant. This simple linear relationship between risk aversion and risk tolerance is defined down below.
RA = 10 - RT
Because we defined RT to range between zero and ten [0, 10], we see that RA will also share that same range - but with the opposite interpretation. An investor with RA of 0 shows no aversion to risk and is often called risk "neutral." Conversely, an individual with RA of 10 is highly risk averse and often chooses a portfolio with limited expected losses.
Our Expected Utility
Here we introduce a new concept called portfolio utility (Up). One might think that there is a direct correlation (1:1) between the expected returns (Rp) from our portfolio and the expected risk of losses. Research has shown that we suffer much greater "pain" from our financial losses than we do "pleasure" from our financial gains. In the relation shown below, the notation of E(*) describes expected values. This may also apply to actual values.
E(Up) = E(Rp) - 1/2*RA*E(Sp)2
We see that our portfolio utility is directly related to our portfolio returns, but that it is diminished by our risk aversion and the variation of these gains (Sp). A higher variation in gains has no effect on the investor with no risk aversion, but a high "penalty" to the utility of a very risk averse individual. Another idea to contemplate is that with our chosen scaling, an "average" investor will have a RA of 5 and the RAs may be displayed in a normal (bell curve) distribution - again ranging from [0, 10].
For the footballer on the pitch (yes, I have been watching Ted Lasso on Apple TV+), their utility describes the amount of pleasure they derive from scoring a goal - offset by the risk of getting kicked in the knee when weaving past their opponent.
Optimal Investment Portfolios
The selection of optimal investment portfolios involves maximizing the expected utility for each individual investor. The outcome of the selection process is a list of investment securities and their respective allocation weights - often graphically displayed as a pie chart. The inputs [E(Rp), RA, and E(Sp)] are noisy and notoriously squirrely to estimate. This is because there is limited historical data with which to model all possible future scenarios. There is the industry-wide boilerplate warning that "past performance is no guarantee of future results." Unfortunately, the only theories we have are derived from observations in our past. As Sir John Templeton once remarked, the four most dangerous words for any investor are: "this time is different." While the data and circumstances may indeed be different, the human response is virtually identical and highly predictable. Think with your brain!